# Black-Scholes-Merton P.D.E

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• Black-Scholes-Merton P.D.E.

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1                                                                       2

Assumptions                                   Black-Scholes-Merton P.D.E.

• One riskless asset(bond), one risky asset(stock) and
a derivative on the stock.
• Risk-free interest rate is constant for every period.
• Stock price follows a geometric Brownian motion.
• Short selling of securities is not limited.
• Neither transaction costs nor taxes.
• All securities are perfectly divisible.
• No dividends from assets during the life of the
derivative.
• No riskless arbitrage opportunities.

3                                                                       4

Black-Scholes-Merton formula                                                                      Figure: B-S-M formula

Price of call option
60

50

40

30

20

10

0
1                        50                     9
price of underlying asset
f(x,t) (T-t = 5)     f(x,t) (T-t = 0.5)   h(x)

5                                                                       6
Deriviation of B-S-M P.D.E.(1)                                Figure: Sample path of SDE
A sample path of S1(t), and Nikkei225 from Nov. 11,1999 to Dec. 12, 2000
(normalized: Nov. 11,1999 = 100)

Ê     Ð
ƒ= 0, ƒ= 0.244(/ year)
250                                                                          Estimated daily
log-return
200
Ave.: 0.00%
150                                                                           Std.: 24.4%
(per annum)
100

50

0

1
54
107
160
213
266
319
372
425
478
531
584
637
690
743
796
849
902
955
a sample path of S(t)          NIKEI2 25(normalized:Nov.11,1999=100)
7                                                                                               8

Deriviation of B-S-M P.D.E.(2)                              Deriviation of B-S-M P.D.E.(3)

9                                                                                              10

Deriviation of B-S-M P.D.E.(4)                              Deriviation of B-S-M P.D.E.(5)

11                                                                                              12
Feynman-Kac formula                                                     Feynman-Kac formula

13                                                                                          14

Stochastic control                                                              Stochastic control

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Stochastic control                                                                         References
•   Black, F., Scholes M., ``The Pricing of Options and Corporate Liabilities",
Journal of Political Economy, 81, 637-659, 1973.
•   Merton, R. C., ``Theory of Rational Option Pricing", Bell Journal of
Economics and Management Science, 4, 141-183, 1973.
•   Duffie, D., Dynamic Asset Pricing Theory, 3rd .ed. Princeton, 2001.

•   “¡cŠx•F•uƒtƒ@Cƒiƒ“X‚ÌŠm—¦‰ð•Í“ü–åv•u’kŽÐƒTƒCƒG“eƒBƒtƒBƒbƒN                (2002)

17                                                                                          18

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